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  <div class="section" id="numpy-linalg-multi-dot">
<h1>numpy.linalg.multi_dot<a class="headerlink" href="#numpy-linalg-multi-dot" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.linalg.multi_dot">
<code class="sig-prename descclassname">numpy.linalg.</code><code class="sig-name descname">multi_dot</code><span class="sig-paren">(</span><em class="sig-param">arrays</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/numpy/numpy/blob/v1.18.1/numpy/linalg/linalg.py#L2566-L2675"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numpy.linalg.multi_dot" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute the dot product of two or more arrays in a single function call,
while automatically selecting the fastest evaluation order.</p>
<p><a class="reference internal" href="#numpy.linalg.multi_dot" title="numpy.linalg.multi_dot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">multi_dot</span></code></a> chains <a class="reference internal" href="numpy.dot.html#numpy.dot" title="numpy.dot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.dot</span></code></a> and uses optimal parenthesization
of the matrices <a class="reference internal" href="#r451bed364cc6-1" id="id1">[1]</a> <a class="reference internal" href="#r451bed364cc6-2" id="id2">[2]</a>. Depending on the shapes of the matrices,
this can speed up the multiplication a lot.</p>
<p>If the first argument is 1-D it is treated as a row vector.
If the last argument is 1-D it is treated as a column vector.
The other arguments must be 2-D.</p>
<p>Think of <a class="reference internal" href="#numpy.linalg.multi_dot" title="numpy.linalg.multi_dot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">multi_dot</span></code></a> as:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">multi_dot</span><span class="p">(</span><span class="n">arrays</span><span class="p">):</span> <span class="k">return</span> <span class="n">functools</span><span class="o">.</span><span class="n">reduce</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">,</span> <span class="n">arrays</span><span class="p">)</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>arrays</strong><span class="classifier">sequence of array_like</span></dt><dd><p>If the first argument is 1-D it is treated as row vector.
If the last argument is 1-D it is treated as column vector.
The other arguments must be 2-D.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>output</strong><span class="classifier">ndarray</span></dt><dd><p>Returns the dot product of the supplied arrays.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><code class="xref py py-obj docutils literal notranslate"><span class="pre">dot</span></code></dt><dd><p>dot multiplication with two arguments.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The cost for a matrix multiplication can be calculated with the
following function:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">cost</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
    <span class="k">return</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
</pre></div>
</div>
<p>Assume we have three matrices
<img class="math" src="../../_images/math/a6e603ba6022b4ebca0f523eb6577a7025b8befe.svg" alt="A_{10x100}, B_{100x5}, C_{5x50}"/>.</p>
<p>The costs for the two different parenthesizations are as follows:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">cost</span><span class="p">((</span><span class="n">AB</span><span class="p">)</span><span class="n">C</span><span class="p">)</span> <span class="o">=</span> <span class="mi">10</span><span class="o">*</span><span class="mi">100</span><span class="o">*</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">10</span><span class="o">*</span><span class="mi">5</span><span class="o">*</span><span class="mi">50</span>   <span class="o">=</span> <span class="mi">5000</span> <span class="o">+</span> <span class="mi">2500</span>   <span class="o">=</span> <span class="mi">7500</span>
<span class="n">cost</span><span class="p">(</span><span class="n">A</span><span class="p">(</span><span class="n">BC</span><span class="p">))</span> <span class="o">=</span> <span class="mi">10</span><span class="o">*</span><span class="mi">100</span><span class="o">*</span><span class="mi">50</span> <span class="o">+</span> <span class="mi">100</span><span class="o">*</span><span class="mi">5</span><span class="o">*</span><span class="mi">50</span> <span class="o">=</span> <span class="mi">50000</span> <span class="o">+</span> <span class="mi">25000</span> <span class="o">=</span> <span class="mi">75000</span>
</pre></div>
</div>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r451bed364cc6-1"><span class="brackets"><a class="fn-backref" href="#id1">1</a></span></dt>
<dd><p>Cormen, “Introduction to Algorithms”, Chapter 15.2, p. 370-378</p>
</dd>
<dt class="label" id="r451bed364cc6-2"><span class="brackets"><a class="fn-backref" href="#id2">2</a></span></dt>
<dd><p><a class="reference external" href="https://en.wikipedia.org/wiki/Matrix_chain_multiplication">https://en.wikipedia.org/wiki/Matrix_chain_multiplication</a></p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p><a class="reference internal" href="#numpy.linalg.multi_dot" title="numpy.linalg.multi_dot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">multi_dot</span></code></a> allows you to write:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">numpy.linalg</span> <span class="kn">import</span> <span class="n">multi_dot</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Prepare some data</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">((</span><span class="mi">10000</span><span class="p">,</span> <span class="mi">100</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">((</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1000</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">((</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">((</span><span class="mi">5</span><span class="p">,</span> <span class="mi">333</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># the actual dot multiplication</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span> <span class="o">=</span> <span class="n">multi_dot</span><span class="p">([</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">])</span>
</pre></div>
</div>
<p>instead of:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">_</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">),</span> <span class="n">C</span><span class="p">),</span> <span class="n">D</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># or</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">B</span><span class="p">)</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">C</span><span class="p">)</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>

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